Fuzzy relational equation with defuzzification - algorithm for the largest solution

This paper provides an algorithm for solving a new fuzzy relational equation including defuzzification. An input fuzzy set is first transformed into an internal fuzzy set by a fuzzy relation, and then an element giving the largest membership value is selected from the support set as an output. Our purpose is to obtain the internal fuzzy relation from pairs of input fuzzy sets and output set-elements. These relational equations are classified into two types, called Type I and Type II in the text, depending on where internal fuzzy sets are defuzzified. Discussions in this paper are done on the largest solutions of these problems. Algorithms to obtain the largest solutions are shown as well as some propositions and numerical examples. (C) 2001 Elsevier Science B.V. All rights reserved.